A098409 Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).

1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, 10153775, 67313377, 448691985, 3004182349, 20188647185, 136094684907, 919884469275, 6232016686995, 42305974804575, 287706424085745, 1959685788407025, 13367193276457881, 91295551930615005, 624255065007468207

Offset: 0

Author

Paul Barry, Sep 07 2004

Keywords

• easy
• nonn

Comments

Binomial transform of A081671. 3rd binomial transform of A000984. Binomial transform is A098410.

Largest coefficient of (1+5*x+x^2)^n; row sums of triangle in A126331. - Philippe Deléham, Oct 02 2007

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in five colors. - N-E. Fahssi, Feb 05 2008

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have five colors. - N-E. Fahssi, Mar 31 2008

Diagonal of rational function 1/(1 - (x^2 + 5*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Isaac DeJager, Madeleine Naquin and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
• Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012 (the sequence b_{5,n}).
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Column 5 of A292627. Cf. A182401.

Programs

Mathematica

Table[SeriesCoefficient[1/(Sqrt[1-3*x]*Sqrt[1-7*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
CoefficientList[Series[1/(Sqrt[1-3x] Sqrt[1-7x]),{x,0,30}],x] (* Harvey P. Dale, Jun 20 2015 *)

Maxima

a(n):=coeff(expand((1+5*x+x^2)^n),x^n);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

PARI

x='x+O('x^66); Vec(1/(sqrt(1-3*x)*sqrt(1-7*x))) \\ Joerg Arndt, May 11 2013

PARI

{a(n) = sum(k=0, n, 7^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

PARI

{a(n) = sum(k=0, n\2, 5^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

Formula

G.f.: 1/sqrt(1-10*x+21*x^2).

E.g.f.: exp(5x)*BesselI(0, 2x).

a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n,k)*binomial(2k,k). - Paul Barry, Mar 08 2005

a(n) = [x^n] (1+5*x+x^2)^n. - Emanuele Munarini, Apr 27 2012

D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012

a(n) ~ 7^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

a(n) = Sum_{k=0..n} 7^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019

a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019

From Peter Bala, Jan 10 2022: (Start)

exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 5*x + 26*x^2 + 140*x^3 + 777*x^4 + ... is the o.g.f. of A182401.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.

a(n) = (1/Pi) * Integral_{x = -1..1} (3 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (7 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)

a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025